Free axial and torsional vibration of boron nitride nanorods using nonlocal finite element method

[1]          A. C. Eringen, D. Edelen, On nonlocal elasticity, International journal of engineering science, Vol. 10, No. 3, pp. 233-248, 1972.

[2]          C. Lim, G. Zhang, J. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, Vol. 78, pp. 298-313, 2015.

[3]          F. Yang, A. Chong, D. C. C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International journal of solids and structures, Vol. 39, No. 10, pp. 2731-2743, 2002.

[4]          A. C. Eringen, Plane waves in nonlocal micropolar elasticity, International Journal of Engineering Science, Vol. 22, No. 8-10, pp. 1113-1121, 1984.

[5]          A. C. Eringen, Nonlocal polar elastic continua, International journal of engineering science, Vol. 10, No. 1, pp. 1-16, 1972.

[6]          A. C. Eringen, Theory of thermomicrofluids, Journal of Mathematical analysis and Applications, Vol. 38, No. 2, pp. 480-496, 1972.

[7]          Y. Zhang, G. Liu, J. Wang, Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression, Physical Review B—Condensed Matter and Materials Physics, Vol. 70, No. 20, pp. 205430, 2004.

[8]          Y. Zhang, G. Liu, X. Xie, Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity, Physical Review B—Condensed Matter and Materials Physics, Vol. 71, No. 19, pp. 195404, 2005.

[9]          Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, Journal of Applied physics, Vol. 98, No. 12, 2005.

[10]        L. Wang, H. Hu, Flexural wave propagation in single-walled carbon nanotubes, Physical Review B—Condensed Matter and Materials Physics, Vol. 71, No. 19, pp. 195412, 2005.

[11]        Ö. Civalek, Ç. Demir, B. Akgöz, Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory, International Journal of Engineering and Applied Sciences, Vol. 1, No. 2, pp. 47-56, 2009.

[12]        O. Civalek, C. Demir, Buckling and bending analyses of cantilever carbon nanotubes using the Euler-Bernoulli beam theory based on non-local continuum model; technical note, 2011.

[13]        C. Wang, Y. Zhang, S. S. Ramesh, S. Kitipornchai, Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory, Journal of Physics D: Applied Physics, Vol. 39, No. 17, pp. 3904, 2006.

[14]        T. Aksencer, M. Aydogdu, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 4, pp. 954-959, 2011.

[15]        M. Aydogdu, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mechanics Research Communications, Vol. 43, pp. 34-40, 2012.

[16]        M. Arda, M. Aydogdu, Torsional statics and dynamics of nanotubes embedded in an elastic medium, Composite Structures, Vol. 114, pp. 80-91, 2014.

[17]        M. Aydogdu, M. Arda, Torsional vibration analysis of double walled carbon nanotubes using nonlocal elasticity, International Journal of Mechanics and Materials in Design, Vol. 12, No. 1, pp. 71-84, 2016.

[18]        M. Şimşek, Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods, Computational Materials Science, Vol. 61, pp. 257-265, 2012.

[19]        K. Mercan, Ö. Civalek, Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Composites Part B: Engineering, Vol. 114, pp. 34-45, 2017.

[20]        B. Uzun, H. Numanoglu, O. Civalek, Free vibration analysis of BNNT with different cross-Sections via nonlocal FEM, Journal of Computational Applied Mechanics, Vol. 49, No. 2, pp. 252-260, 2018.

[21]        H. Numanoglu, K. Mercan, Ö. Civalek, Finite element model and size-dependent stability analysis of boron nitride and silicon carbide nanowires/nanotubes, Scientia Iranica, Vol. 26, No. 4, 2019.

[22]        A. O. Maselugbo, H. B. Harrison, J. R. Alston, Boron nitride nanotubes: a review of recent progress on purification methods and techniques, Journal of Materials Research, Vol. 37, No. 24, pp. 4438-4458, 2022.

[23]        H. Xu, Q. Wang, G. Fan, X. Chu, Theoretical study of boron nitride nanotubes as drug delivery vehicles of some anticancer drugs, Theoretical Chemistry Accounts, Vol. 137, No. 7, pp. 104, 2018.

[24]        Z. K. Maraghi, A. G. Arani, O. Civalek, Comparative analysis of carbon and boron-nitride nanotube reinforcements on the vibration characteristics of magnetostrictive sandwich plates, Composite Structures, Vol. 361, pp. 119029, 2025.

[25]        A. Tepe, Comparative size-dependent free vibration analysis of boron nitride and silicon carbide nanotubes based on strain gradient Euler–Bernoulli beam theory, Mechanics Based Design of Structures and Machines, pp. 1-21, 2025.

[26]        J. Phadikar, S. Pradhan, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational materials science, Vol. 49, No. 3, pp. 492-499, 2010.

[27]        H. M. Numanoğlu, Ö. Civalek, On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM, International Journal of Mechanical Sciences, Vol. 161, pp. 105076, 2019.

[28]        A. Yazıcıoğlu, Ö. Civalek, Axial Vibration of a Nanoring Rod Using Nonlocal Finite Element Method, International Journal of Engineering and Applied Sciences, Vol. 17, No. 2, pp. 73-80, 2025.

[29]        M. Eltaher, S. A. Emam, F. Mahmoud, Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation, Vol. 218, No. 14, pp. 7406-7420, 2012.

[30]        M.-O. Belarbi, S. Benounas, L. Li, P. Van Vinh, A. Garg, Vibration of a functionally graded doubly curved shallow nanoshell: an improved FSDT model and its nonlocal finite element implement, Journal of Vibration Engineering & Technologies, Vol. 13, No. 1, pp. 93, 2025.

[31]        Ö. Civalek, B. Uzun, M. Ö. Yaylı, B. Akgöz, Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method, The European Physical Journal Plus, Vol. 135, No. 4, pp. 381, 2020.

[32]        V.-K. Tran, Q.-H. Pham, T. Nguyen-Thoi, A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations, Engineering with Computers, Vol. 38, No. 2, pp. 1465-1490, 2022.

[33]        B. Uzun, Ö. Civalek, M. Ö. Yaylı, Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions, Mechanics Based Design of Structures and Machines, Vol. 51, No. 1, pp. 481-500, 2023.

[34]        Y. Tang, P. Bian, H. Qing, Finite element formulation for free vibration of the functionally graded curved nonlocal nanobeam resting on nonlocal elastic foundation, Journal of Vibration and Control, pp. 10775463241278642, 2024.

[35]        Ö. Civalek, H. M. Numanoğlu, Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods, International Journal of Mechanical Sciences, Vol. 188, pp. 105939, 2020.

[36]        A. S. Kumar, S. Kumar, P. K. Choudhary, A. Gupta, A. Narayan, Free vibration characteristics of elastic foundation-supported porous functionally graded nanoplates using Rayleigh-Ritz approach, International Journal of Structural Integrity, Vol. 15, No. 2, pp. 298-321, 2024.

[37]        Ö. E. Genel, H. Koç, E. Tüfekci, In-Plane Static Analysis of Curved Nanobeams Using Exact-Solution-Based Finite Element Formulation, Computers, Materials & Continua, Vol. 82, No. 2, 2025.

[38]        Q.-H. Pham, V. K. Tran, T. T. Tran, V. C. Nguyen, A. M. Zenkour, Nonlocal higher-order finite element modeling for vibration analysis of viscoelastic orthotropic nanoplates resting on variable viscoelastic foundation, Composite Structures, Vol. 318, pp. 117067, 2023.

[39]        H. Koc, E. Tufekci, A novel approach of bending behavior of carbon nanotubes by combining the effects of higher-order boundary conditions and coupling through doublet mechanics, Mechanics of Advanced Materials and Structures, Vol. 31, No. 27, pp. 8905-8919, 2024.

[40]        E. Yıldırım, I. Esen, Effect of the porous structure on the hygrothermal vibration analysis of functional graded nanoplates using nonlocal high-order continuum plate model, Acta Mechanica, Vol. 235, No. 8, pp. 5079-5106, 2024.

[41]        A. E. Abouelregal, Ö. Civalek, B. Akgöz, A size-dependent non-fourier heat conduction model for magneto-thermoelastic vibration response of nanosystems, Journal of Applied and Computational Mechanics, Vol. 11, No. 2, pp. 344-357, 2025.

[42]        D. M. Lan, N. D. Anh, P. V. Dong, D. V. Thom, Ö. Civalek, P. V. Minh, A new Galerkin method for buckling of sandwich nanobeams, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 239, No. 8, pp. 3034-3051, 2025.

[43]        M. S. Vaccaro, R. Luciano, F. Marotti de Sciarra, Size-dependent stress-driven behaviour of nanobeams based on higher-order theories, Meccanica, pp. 1-22, 2025.

[44]        M. Akpınar, B. Uzun, H. G. Kadıoğlu, M. Ö. Yaylı, Dynamics of a Short-Fiber-Reinforced Nanorod via Hardening Nonlocal Approach, Mechanics of Solids, pp. 1-23, 2025.

[45]        G. La Valle, C. Soize, A higher-order nonlocal elasticity continuum model for deterministic and stochastic particle-based materials, Zeitschrift für angewandte Mathematik und Physik, Vol. 75, No. 2, pp. 49, 2024.

[46]        R. Barretta, R. Luciano, F. M. de Sciarra, M. S. Vaccaro, Modelling issues and advances in nonlocal beams mechanics, International Journal of Engineering Science, Vol. 198, pp. 104042, 2024.

[47]        A. Tariq, B. Uzun, B. Deliktaş, M. Ö. Yaylı, Application of machine learning methodology for investigating the vibration behavior of functionally graded porous nanobeams, The Journal of Strain Analysis for Engineering Design, Vol. 60, No. 2, pp. 131-151, 2025.

[48]        Ş. D. Akbaş, S. Dastjerdi, B. Akgöz, Ö. Civalek, Dynamic analysis of functionally graded porous microbeams under moving load, Transport in Porous Media, Vol. 142, No. 1, pp. 209-227, 2022.

[49]        S. Dastjerdi, Ö. Civalek, M. Malikan, B. Akgöz, On analysis of nanocomposite conical structures, International Journal of Engineering Science, Vol. 191, pp. 103918, 2023.

[50]        P. Van Vinh, A novel modified nonlocal strain gradient theory for comprehensive analysis of functionally graded nanoplates, Acta Mechanica, Vol. 236, No. 1, pp. 173-204, 2025.

[51]        Ö. Civalek, M. Akpınar, B. Uzun, M. Ö. Yaylı, Dynamics of a non-circular-shaped nanorod with deformable boundaries based on second-order strain gradient theory, Archive of Applied Mechanics, Vol. 94, No. 11, pp. 3555-3572, 2024.

[52]        M. Mehdi, Buckling and bending analyses of a sandwich beam based on nonlocal stress-strain elasticity theory with porous core and functionally graded facesheets, Advances in materials research: AMR, Vol. 11, No. 4, pp. 279-298, 2022.

[53]        A. E. Abouelregal, B. Akgöz, Ö. Civalek, Nonlocal thermoelastic vibration of a solid medium subjected to a pulsed heat flux via Caputo–Fabrizio fractional derivative heat conduction, Applied Physics A, Vol. 128, No. 8, pp. 660, 2022.

[54]        M. Mahdavi Adeli, R. Mahmoudi, A. Soleimani, Pull-in behaviour of a micro switch actuated by the electrostatic under a uniform longitudinal magnetic field based on nonlocal couple stress theory, Journal of Computational Applied Mechanics, Vol. 54, No. 4, pp. 577-587, 2023.

[55]        M. A. Mohammadimehr, A. Loghman, S. Amir, M. Mohammadimehr, E. Arshid, Ö. Civalek, Magneto-electro vibration analysis of a moderately thick double-curved sandwich panel with porous core and GPLRC using FSDT, Journal of Computational Applied Mechanics, Vol. 56, No. 3, pp. 673-693, 2025.

[56]        M. Nejadi, M. Mohammadimehr, M. Mehrabi, Free vibration and stability analysis of sandwich pipe by considering porosity and graphene platelet effects on conveying fluid flow, Alexandria Engineering Journal, Vol. 60, No. 1, pp. 1945-1954, 2021.

[57]        A. Alibakhshi, S. Dastjerdi, B. Akgöz, Ö. Civalek, Parametric vibration of a dielectric elastomer microbeam resonator based on a hyperelastic cosserat continuum model, Composite Structures, Vol. 287, pp. 115386, 2022.

[58]        M. Mohammadimehr, The effect of a nonlocal stress-strain elasticity theory on the vibration analysis of Timoshenko sandwich beam theory, Advances in nano research, Vol. 17, No. 3, pp. 275-284, 2024.

[59]        A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,  pp. 1983.

[60]        N. G. Chopra, A. Zettl, Measurement of the elastic modulus of a multi-wall boron nitride nanotube, Solid State Communications, Vol. 105, No. 5, pp. 297-300, 1998.

[61]        L. Jiang, W. Guo, A molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes, Journal of the Mechanics and Physics of Solids, Vol. 59, No. 6, pp. 1204-1213, 2011.

[62]        U. Uyan, Yerel Olmayan Elastisite Teorisi İle Bor Nitrür Nanoçubukların Eksenel Titreşimi, Master Degree Thesis, Civil Engeenering, Akdeniz Univercity, Turkey, 2018.